Prediction of Ue3 and \cos 2\theta_23 from discrete symmetry

We discuss the question why the mixing $U_{e3}$ is small. The natural answer is $U_{e3}=0$ in some symmetric limit, in which two large mixings are realized. It is possible to force $U_{e3}$ and $\cos 2{\theta }_{23}$ to be zero by imposing a discrete symmetry. We investigate a special class of symmetries $Z_2$ and of the consequences of their perturbative violation.